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7. A quart contains 1600 beans of average size, and a field is planted with 22 rows of 800 hills each, with 6 beans in a hill. The increase is tenfold. What is the value of the crop at $3 a bushel? (There are 32 quarts in a bushel.)

8. For making 25 gallons of ordinary beer 60 pounds of barley and 0.5 of a pound of hops are needed. If the barley costs $1.50 for 60 pounds, and the hops cost 18 cents a pound, what is the profit to the brewer on a cask of 42 gallons if he sells it for $5 and reckons his labor $1.50?

9. A person receives his income quarterly.

The first

quarter he receives $533.25, the second $1535.20, the third $856.44, the fourth $ 725.19. His expenses for these quarters are respectively $686.60, $ 734.25, $589.15, $849.65. How much does he save for the year?

10. A hen lays on an average 120 eggs a year worth 24 cents a dozen. She eats a quart of barley every 5 days. The barley is worth 56 cents a bushel (32 quarts). What is the annual profit from this hen?

11. A square garden measuring on each side 40.50 yards is enclosed by three lines of galvanized iron wire. Eight yards of this wire weigh a pound, and it is worth 7.5 cents per pound. What is the cost of the wire?

12. A family composed of five persons consumes daily one pound of stale bread for each person, or 1.15 pounds of fresh bread. If bread is worth 5 cents a pound, find the annual saving which this family will make if it eats stale bread altogether.

CHAPTER VII.

MULTIPLES AND MEASURES.

85. When the multiplier is an integral number, the product is called a multiple of the multiplicand; and, in division, when the quotient is an integral number, the divisor is called a measure of the dividend. Thus, 8 × 7=56; the number 56 is a multiple of 7. Again, 56÷ 7 = 8; the number 7 is a measure of 56.

86. A number which cannot be divided by any other number except unity without remainder is called a prime number.

Thus, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc., are prime numbers.

87.

Other numbers are each the product of a fixed set of prime numbers, and are called composite numbers.

88. Numbers which can be divided by 2 without remainder are called even numbers; and all other numbers are called odd numbers. Even numbers end in 2, 4, 6, 8, or 0; odd numbers end in 1, 3, 5, 7, or 9.

89. By way of distinction, when a number is used without reference to any designated unit, it is called an abstract number; and, when used with reference to a specified unit, it is called a concrete number.

Thus, 5, 7, 8 are abstract numbers, and 5 horses, 7 chairs, 8 dollars are called, by way of distinction, concrete numbers.

90. To factor a composite number is to separate the number into its factors.

Find the prime factors of 144.

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That is, 1442×2× 2 × 2 × 3 × 3.

91. To avoid the necessity of writing long rows of equal factors, a small figure called the exponent is written at the right of a number to show how many times the number is taken as a factor.

Thus, 2 × 2 × 2 × 2 × 3 × 3 is written 2a × 32.

The expression 24 is called the fourth power of 2, and 32 is called the second power of 3.

92. It is evident, from the above example, that the method of separating a composite number into its prime factors is,

Divide the given number by any prime number that is contained in it without remainder; then the quotient by any prime number that is contained in it without remainder; and so on until the quotient is itself a prime number. The several divisors and the last quotient are the prime factors. If no prime factor is found before the quotient becomes equal to or less than the divisor, the number is prime.

93. The following tests are useful for determining without actual division if a number contains certain factors:

1. A number is divisible by 2 if its last digit is even. 2. A number is divisible by 4 (22) if the number denoted by the last two digits is divisible by 4.

3. A number is divisible by 8 (23) if the number denoted by the last three digits is divisible by 8.

4. A number is divisible by 3 if the sum of its digits is divisible by 3.

5. A number is divisible by 9 (32) if the sum of its digits is divisible by 9.

6. A number is divisible by 5 if its last digit is either 5 or 0.

7. A number is divisible by 25 (5) if the number denoted by the last two digits is divisible by 25.

8. A number is divisible by 125 (53) if the number denoted by the last three digits is divisible by 125.

9. A number is divisible by 6 if its last digit is even, and the sum of its digits is divisible by 3.

10. A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either O or a multiple of 11.

Ex. 61.

Find the prime factors of:

1. 32; 48; 56; 60; 75; 63; 92; 44; 88; 72; 84; 85. 2. 51; 69; 68; 87; 54; 98; 74; 90; 86; 70; 42; 62. 3. 112; 140; 132; 216; 162; 176; 252; 240; 360; 384. 4. 484; 476; 512; 525; 560; 572; 632; 648; 696; 720. 5. 748; 775; 824; 876; 888; 948; 960; 925; 117; 119.

94. The number 1.56 may be put in the form of 156 × .01, and thus separated into 22 x 3 x 13 x .01.

Ex. 62.

Find the prime factors of:

1. 1.05; 12.5; 14.3; 1.65; 19.2; 2.42; 62.4; 27.5.

2. 34.3; 5.39; 62.1; 118.8; 1.331; 1.452; 1.584; 92.4.

GREATEST COMMON MEASURE.

95. The measures of 12 are 1, 2, 3, 4, 6, 12, and the measures of 18 are 1, 2, 3, 6, 9, 18. These two numbers have the measures 1, 2, 3, 6 in common, and of these measures 6 is the greatest.

The measures that two or more numbers have in common are called their common measures, and the greatest of these is called their Greatest Common Measure, which, for the sake of brevity, is denoted by the letters G. C. M.

If two or more numbers have no common measure they are said to be prime to each other. Thus, 27 and 125 are prime to each other.

96. The prime factors of 12 are 22, 3.

The prime factors of 18 are 2, 32.

The prime factors common to 12 and 18 are 2, 3. The G. C. M. of 12 and 18, namely 6, is 2 × 3.

That is, the G. C. M. of two or more numbers is,

The product of the prime factors common to the numbers, each prime factor having the least exponent that it has in any one of the numbers.

Hence, to find the G. C. M. of two or more numbers,
Separate the numbers into their prime factors.

Select the lowest power of each factor that is common to the given numbers, and find the product of these powers. Find the G. C. M. of 84, 105, 63.

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